Integer Programming Duality Research Articles

Wiley encyclopedia of operations research and management science

Use of Lagrange Interpolating Polynomials in RLT. Systems in Series. Parallel Configurations. Spanning trees. Recycling. Split Cuts. Retrial Queues. Horse Racing. MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique). Dual simplex. History of CP. Procurement Contracts. MRP. Just in Time. Push and Pull Production. MULTIVARIATE INPUT MODELING. Operational Research Society. The M/G/1 Queue. Drama Theory. Hyper-heuristics. Antithetic Variates. Music. Edgeworth market games. Operational Risk. Queueing Notation. Environmental impact. Why variance is not risk. Conjugate gradient methods. Birth-and-Death Processes. Surgery Planning and Scheduling. Large-scale linear models. Standby Systems. Billiards/Pool. AXIOMATIC MEASURES OF RISK AND RISK-VALUE MODELS. Age Replacement Policies. PASTA and Related Results. Collaborative Procurement. LP projection algorithms. R&D risk management. Bioterrorism. Vendor-Managed Inventory. Gomory Cuts. Travel demand modeling. System Availability. Infinite horizon problems. Trust. Bayesian Network Classifiers. Integer Programming Duality. Queueing Disciplines. Stakeholder participation. Models and Basic Properties. Parametric LP analysis. Operations Research and Golfing. Ice Hockey. Clustering. Finite-Population Models. Batch Arrivals and Service. Optimal Monitoring Strategies. Basic Polyhedral Theory. Operations Research and Art. Managing portfolios of risks. Measures of risk equity. Branch and Bound Algorithms. Analytics in Retail. Variational Inequalities. Cross-Entropy Method. TSP heuristics. Capacity allocation. Control Variates. Cover Inequalities. Direct Search Methods. NLP Software. Bilinear optimization. Discrete-Time Martingales. Branch and Price. Column generation. IP Preprocessing. Optimal Reliability Allocation. Simplex-based LP solvers. Instance Formats. Clique generalizations. Availability in Stochastic Models. Evolutionary Algorithms. Basic CP theory: Search. Multimethodology. Branch and Cut. Memetic Algorithms. Urban mass transit. Coherent Systems. Subgradient optimization. Klimov's Model. Credit risk assessment. Branch-width and Tangles. Stochastic Hazard Process. Deterministic Global Optimization. Multistage (stochastic) games. Basis Reduction Methods. ACCIDENT PRECURSORS AND WARNING SYSTEMS MANAGEMENT: A BAYESIAN APPROACH TO MATHEMATICAL MODELS. Genetic Algorithms. Newsvendor models. Parallel Systems. Opportunities in Tennis. Job-shop scheduling. Simplex method and complexity. Geometric programming. Evacuation Planning. The Performance? Allocation Games. Scenario Generation. Robustness Analysis. Dynamic auctions. Neuroeconomics and game theory. Common Random Numbers. Selective Support Vector Machines. Semi-Markov Processes. Risk Averse Models. Sampling Methods. Domination problems. Simulation of Rare Events. Introduction to Tabu Search. Basic Interdiction Models. Introductory Concepts. Remanufacturing. PAUSE procedure. Wardrop equilibria. Mass customization. Ellipsoidal algorithms. Benders decomposition. Dynamic vehicle routing. Warranty Modeling. Design for Network Resiliency. Passenger rail transportation. Vehicle Routing Problem. Supply Chain Outsourcing. Lovasz-Schriver Reformulation. Hazard Rate Function. Lot-sizing. Differential games. Regenerative Processes. Guided Local Search. Preferences for repeated gambles. Percolation Theory. Simulated Annealing. Scoring Rules. Mixing Sets. Cricket. Prospect Theory. Combining Forecasts. Graph search techniques. Quasi-Newton methods. Random search algorithms. Supply chain coordination. Service Outsourcing. Markov Renewal Processes. K-out-of-n Systems.

Non-linear anonymous pricing combinatorial auctions

In combinatorial auctions the pricing problem is of main concern since it is the means by which the auctioneer signals the result of the auction to the participants. In order for the auction to be regarded as fair among the various participants the price signals should be such that a participant that has won a subset of items knows why his bid was a winning bid and that agents that have not acquired any item easily can detect why they lost. The problem in the combinatorial auction setting is that the winner determination problem is a hard integer programming problem and hence a linear pricing scheme supporting the optimal allocation might not exist. From integer programming duality theory we know that there exist non-linear anonymous price functions that support the optimal allocation. In this paper we will provide a means to obtain a simple form of a price system that supports the optimal allocation. Our method relies on the fact that we separate the solution of the winner determination problem and the pricing problem. This separation yields a non-linear price function of a much simpler form compared to when the two problems are solved simultaneously. The pure pricing problem is formulated as a mixed-integer program. The procedure is computationally tested using difficult instances of the combinatorial auctions test suite [16] . The results indicate that the number of extreme prices forming the non-linear anonymous price system is small.

Equilibrium prices supported by dual price functions in markets with non-convexities

The issue of finding market clearing prices in markets with non-convexities has had a renewed interest due to the deregulation of the electricity sector. In the day-ahead electricity market, equilibrium prices are calculated based on bids from generators and consumers. In most of the existing markets, several generation technologies are present, some of which have considerable non-convexities, such as capacity limitations and large start-up costs. In this paper we present equilibrium prices composed of a commodity price and an uplift charge. The prices are based on the generation of a separating valid inequality that supports the optimal resource allocation. In the case when the sub-problem generated as the integer variables are held fixed to their optimal values possess the integrality property, the generated prices are also supported by non-linear price functions that are the basis for integer programming duality.

Non-Linear Anonymous Pricing in Combinatorial Auctions

In combinatorial auctions the pricing problem is of main concern since it is the means by which the auctioneer signals the result of the auction to the participants. In order for the auction to be regarded as fair among the various participants the price signals should be such that a participant that has won a subset of items knows why his bid was a winning bid and that agents that have not acquired any item easily can detect why they lost. The problem in the combinatorial auction setting is that the winner determination problem is a hard integer programming problem and hence in general there does not exist a linear pricing scheme supporting the optimal allocation. This means that single item prices, that support the optimal allocation can in general not be found. In this article we present an alternative. From integer programming duality theory we know that there exist non-linear anonymous price functions that support the optimal allocation. In this paper we will provide a means to obtain a simple form of a non-linear anonymous price system that supports the optimal allocation. Our method relies on the fact that we separate the solution of the winner determination problem and the pricing problem. This separation yields a non-linear price function of a much simpler form compared to when the two problems are solved simultaneously. The pure pricing problem is formulated as a mixed-integer program. If solving this program is too demanding computationally a heuristic can be used which essentially requires us to solve a sequence of linear programming relaxations of a new mixed-integer programming formulation of the pricing problem. The procedure is computationally tested using instances of the combinatorial auctions test suite.

Pricing in combinatorial auctions

Our objective is to find prices on individual items in a combinatorial auction that support the optimal allocation of bundles of items, i.e. the solution to the winner determination problem of the combinatorial auction. The item-prices should price the winning bundles according to the corresponding winning bids, whereas the bundles that do not belong to the winning set should have strictly positive reduced cost. That is the bid on a non-winning bundle is strictly less than the sum of prices of the individual items that belong to the bundle, thus providing information to the bidders why they are not in the winning set. Since the winner determination problem is an integer program, in general we cannot find a linear price-structure with these characteristics. However, integer programming duality can be used to obtain this kind of price-information. Normally, it is computationally too expensive to derive the integer programming dual function, but in an iterative combinatorial auction it might be worth to do it since the information provided to the bidders from the non-linear dual function is of great importance for the bidders. Throughout, the ideas are illustrated by means of numerical examples.

Towards Strong Duality in Integer Programming

We consider in this paper the Lagrangian dual method for solving general integer programming. New properties of Lagrangian duality are derived by a means of perturbation analysis. In particular, a necessary and sufficient condition for a primal optimal solution to be generated by the Lagrangian relaxation is obtained. The solution properties of Lagrangian relaxation problem are studied systematically. To overcome the difficulties caused by duality gap between the primal problem and the dual problem, we introduce an equivalent reformulation for the primal problem via applying a pth power to the constraints. We prove that this reformulation possesses an asymptotic strong duality property. Primal feasibility and primal optimality of the Lagrangian relaxation problems can be achieved in this reformulation when the parameter p is larger than a threshold value, thus ensuring the existence of an optimal primal-dual pair. We further show that duality gap for this partial pth power reformulation is a strictly decreasing function of p in the case of a single constraint.

Efficient market-clearing prices in markets with nonconvexities

This paper addresses the existence of market clearing prices and the economic interpretation of strong duality for integer programs in the economic analysis of markets with nonconvexities (indivisibilities). Electric power markets in which nonconvexities arise from the operating characteristics of generators motivate our analysis; however, the results presented here are general and can be applied to other markets in which nonconvexities are important. We show that the optimal solution to a linear program that solves the mixed integer program has dual variables that: (1) have the traditional economic interpretation as prices; (2) explicitly price integral activities; and (3) clear the market in the presence of nonconvexities. We then show how this methodology can be used to interpret the solutions to nonconvex problems such as the problem discussed by Scarf [Journal of Economic Perspectives 8(4) (1994) 111].

Generating functions and duality for integer programs

We consider the integer program P → max { c ′ x | Ax = y ; x ∈ N n } . Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables λ ∈ R m have discrete analogues z ∈ C m , related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and | z | for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P * , the analogue of the dual LP in linear programming.

Integer Programming Duality in Multiple Objective Programming

The weighted sums approach for linear and convex multiple criteria optimization is well studied. The weights determine a linear function of the criteria approximating a decision makers overall utility. Any efficient solution may be found in this way. This is not the case for multiple criteria integer programming. However, in this case one may apply the more general e-constraint approach, resulting in particular single-criteria integer programming problems to generate efficient solutions. We show how this approach implies a more general, composite utility function of the criteria yielding a unified treatment of multiple criteria optimization with and without integrality constraints. Moreover, any efficient solution can be found using appropriate composite functions. The functions may be generated by the classical solution methods such as cutting plane and branch and bound algorithms.

Exact Penalty Function and Asymptotic Strong Nonlinear Duality in Integer Programming

In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties. The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].

Weyl-minkowski duality for integarl monoids*

The abstract linear duality model of Carvalho and Trotter is extended to accornodate a duality relation shared by point sets and families of (possibly) nonlinear functions. Elementary properties of the general (linear) are established along with results linking finiteness conditions which correspond to well-known (linear) relations of Weyl, Farkas, Minkowski, Lehman and Fulkerson. The particular instance of integer programming duality serves as the main motivation for the general model; duality between finitely generated integral monoids and finite families of Chvatal functions is established in direct analogy to the classical weyl- Minkowski auaiity of iiniteiy generated and polyhedral cones. Affine versions of the iniegrai monoid duality results are developed, i.e., duality relations for the integral elements of rational polyhedra.

The group problem and integer programming duality

Some duality results for integer programming based on subadditive functions are presented first for linear programs and then for the group problem. A similar result for the knapsack problem is given and then a relationship between facets for the group relaxation and facets of the knapsack problem is given. The mixed integer cyclic group problem is then considered and a dual problem given. A common theme is to try to characterize the strongest possible dual problem or equivalently the smallest possible cone of subadditive functions.

Surrogate Dual Multiplier Search Procedures in Integer Programming

Search procedures for optimal Lagrange multipliers are highly developed and provide good bounds in branch and bound procedures that have led to the successful application of Lagrangean duality in integer programming. Although the surrogate dual generally provides a better objective bound, there has been little development of surrogate multiplier search procedures. This paper develops and empirically analyzes several surrogate multiplier search procedures. Results indicate that the procedures can produce possibly superior bounds in an amount of time comparable to other techniques. Our discussion also highlights the similarity of the procedures to some well known Lagrangean search techniques.

Integer programming duality: Price functions and sensitivity analysis

Recently a duality theory for integer programming has been developed. Here we examine some of the economic implications of this theory, in particular the necessity of using price functions in place of prices, and the possibility of carrying out sensitivity analysis of optimal solutions. In addition we consider the form of price functions that are generated by known algorithms for integer programming.

Some aspects of integer programming duality

Two approaches to duality in integer programming are briefly reviewed in this paper with emphasis on their applicability. We introduce the so called integer Lagrangean relaxation for a given problem and show how it can be used in a construction of an appropriate dual problem for which strong duality holds. A branch-and-bound algorithm for solving both primal and dual problems is described. Finally we stress the importance of a reformulation of a given integer problem in seeking its optimal dual variables and their interpretation.

Some relationships between lagrangian and surrogate duality in integer programming

Lagrangian dual approaches have been employed successfully in a number of integer programming situations to provide bounds for branch-and-bound procedures. This paper investigates some relationship between bounds obtained from lagrangian duals and those derived from the lesser known, but theoretically more powerful surrogate duals. A generalization of Geoffrion's integrality property, some complementary slackness relationships between optimal solutions, and some empirical results are presented and used to argue for the relative value of surrogate duals in integer programming. These and other results are then shown to lead naturally to a two-phase algorithm which optimizes first the computationally easier lagrangian dual and then the surrogate dual.

Constructive Duality in Integer Programming

A new integer programming dual problem is constructed from a reformulation of the integer programming problem. Properties of this integer programming dual problem are developed and it is shown that strong integer programming cuts and surrogate constraints can be derived from optimal dual solutions. A primal–dual ascent algorithm for solution of the integer programming dual problem is presented, and the use of this algorithm in branch-and-bound is explained.